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From Huygens odd sympathy to the energy Huygens' extraction from the sea waves


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AACIMP 2011 Summer School. Science of Global Challenges Stream. Lecture by Tomasz Kapitaniak.

AACIMP 2011 Summer School. Science of Global Challenges Stream. Lecture by Tomasz Kapitaniak.

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  • 1. From Huygens odd sympathy to the energy Huygens extraction from the sea waves Tomasz Kapitaniak Technical University of Lodz
  • 2. Co-workers Co workers• Przemysław Perlikowski• Krzysztof Czołczynski• Andrzej Stefański• Marcin Kapitaniak
  • 3. Our publications• K. C., P. P., A. S. and T. K.: "Clustering and synchronization of n Huygens clocks", Physica A 388 (2009)• K. C., P. P., A. S. and T. K.: "Clustering of Huygens clocks", Progress of , , g yg , g Theoretical Physics Vol. 122, No. 4 , (2009)• K. C., P. P., A. S. and T. K.: "Clustering of non-identical clocks", Progress of Theoretical Physics Vol 125 No 3 (2011) Vol.125, No.• K.C., P.P., A.S. and T.K: ‘Why two clocks synchronize: Energy balance of synchronized clocks”, Chaos, 21, 023129 (2011)• K. C., P. P., A. S. and T. K.: "Huygens odd sympathy experiment revisited", International Journal Bifurcation and Chaos (2011) – in press
  • 4. Salisbury Cathedral - 1386
  • 5. John Constable – ca 1825 ca.
  • 6. Pendulum clock conceived by Galileo Galilei around 1637
  • 7. Christiaan Huygensby Bernard Vaillant, Museum Hofwijck, Voorburg
  • 8. The first pendulum clock, invented by Christiaan Huygens in 1656
  • 9. The pendulum clock was invented in 1656 by Dutchscientist Christiaan Huygens, and patented the followingyear. Huygens was inspired by investigations ofpendulums b G lil d l by Galileo G lil i b i i Galilei beginning around 1602 d 1602.Galileo discovered the key property that makespendulums useful timekeepers: isochronism, whichmeans th t th period of swing of a pendulum i that the i d f i f d l isapproximately the same for different sized swings.Galileo had the idea for a pendulum clock in 1637, partlyconstructed b hi son i 1649 b neither li d to fi i h d by his in 1649, but i h lived finishit.The introduction of the pendulum, the first harmonicoscillator used in timekeeping, increased the accuracy of p g yclocks enormously, from about 15 minutes per day to 15seconds per day leading to their rapid spread as existingverge and foliot clocks were retrofitted with pendulums. g p
  • 10. Sketch of basic componentsof a pendulum clock withanchor escapement. (FromMartinek & Rehor 1996.))Parts:- pendulum; p- anchor escapement arms;- escape wheel;- gear train,- gravity-driven weight.
  • 11. Edward East winged lantern clock
  • 12. These early clocks, d t th i verge escapements, h dTh l l k due to their t hadwide pendulum swings of up to 100°. In his 1673analysis of pendulums, Horologium Oscillatorium, y p , g ,Huygens showed that wide swings made the penduluminaccurate, causing its period, and thus the rate of theclock,clock to vary with unavoidable variations in the drivingforce provided by the movement. Clockmakersrealization that only pendulums with small swings of afew degrees are isochronous motivated the invention ofthe anchor escapement around 1670, which reduced thependulum spendulums swing to 4°-6° 4 6
  • 13. Verge escapementVerge escapement showing (c) crown wheel, g p g( )(v) verge rod, (p,q) pallets.
  • 14. It was used in the first mechanical clocks and was originally controlled byfoliot, a horizontal bar with weights at either end. The escapement consistsof an escape wheel shaped some hat like a cro n with pointed teeth heel somewhat crown, ithsticking axially out of the side, oriented vertically. In front of the crownwheel is a vertical shaft, the verge, attached to the foliot at top, whichcarries two metal plates ( ll ) sticking out lik fl i l l (pallets) i ki like flags f from a fl pole, flag loriented about ninety degrees apart, so only one engages the crown wheelteeth at a time. As the wheel turns, one tooth pushes against the upperpallet, rotating the verge and the attached foliot. As the tooth pushes pastthe upper pallet, the lower pallet swings into the path of the teeth on theother side of the wheel. A tooth catches on the lower pallet, rotating the p , gverge back the other way, and the cycle repeats. A disadvantage of theescapement was that each time a tooth lands on a pallet, the momentum ofthe foliot pushes the crown wheel backwards a short distance before theforce of the wheel reverses the motion.
  • 15. Anchor escapement1660 by Robert Hooke
  • 16. Reuleaux-Voight model X-3
  • 17. Escapement mechanism
  • 18. First step 0<ϕi<γN (i=1,2) then MDi=MNi and when ϕ1<0 then MDi=0. p ( )Second stage -γN <ϕ1<0 MDi=-MNi and for ϕ1>0 MDi=0.
  • 19. Works on clock dynamic • Huygens, C. Letters to Father (1665) • Blekham, I.I., “Synchronization in Science and Technology,” (ASME N Y k 1988) T h l ” (ASME, New York,1988). • Bennet, M., et al. “Huygens’s clocks,” Proc. Roy. Soc. London, 458, 563-579 (2002). S L d A 458 563 579 (2002) • Roup, A. et al.: “Limit cycle analysis of the verge and foliot clock escapement using impulsive differential equations and Poincare maps,” Int. J. Control, 76, 1685-1698 (2003). • Moon, F. and Stiefel, P., “Coexisting chaotic and periodic dynamics in clock escapements,” Phil. Trans. R. Soc. A, 364, 2539 (2006).
  • 20. Metronome - clock
  • 21. Shortly after The Royal Society’s founding in 1660, Christiaan Huygens, in partnership with theSociety, set out to solve the outstanding technological challenge of the day: the longitudepproblem, i.e. - finding a robust, accurate method of determining longitude for maritime g g gnavigation (Yoder 1990). Huygens had invented the pendulum clock in 1657 (Burke 1978) and,subsequently, had demonstrated mathematically that a pendulum would follow an isochronouspath, independent of amplitude, if cycloidal-shaped plates were used to confine the pendulumsuspension (Yoder 1990). Huygens believed that cycloidal pendulum clocks, suitably modified towithstand the rigours of sea travel, could provide timing of sufficient accuracy to determinelongitude reliably. Maritime pendulum clocks were constructed by Huygens in collaborationwith one of the original fellows of The Royal Society, Alexander Bruce, 2nd Earl of Kincardine.Over the course of three years (1662-1665) Bruce and the Society supervised sea trials of theclocks. Meanwhile, Huygens, remaining in The Hague, continually corresponded with theSociety thS i t through Sir Robert Moray, both t inquire about the outcome of the sea trials and to h Si R b t M b th to i i b t th t f th ti l dtdescribe the ongoing efforts Huygens was making to perfect the design of maritime clocks. On 1March 1665, Moray read to the Society a letter from Huygens, dated 27 February 1665, reportingof (Birch 1756):an odd kind of sympathy perceived by him in these watches [twomaritime clocks] suspended by the side of each other. ] p y
  • 22. Huygens’s study of two clocks operating simultaneously arose from the practical requirement ofredundancy for maritime clocks: if one clock stopped (or had to be cleaned), then the other couldbe used to provide timekeeping ( yg p p g (Huygens 1669). In a contemporaneous letter to his father, ) pHuygens further described his observations made while confined to his rooms by a brief illness.Huygens found that the pendulum clocks swung in exactly the same frequency and 180o out ofphase (Huygens 1950a; b). When he disturbed one pendulum, the anti-phase state was restoredwithin half an hour and remained indefinitely. Motivated by the belief that synchronization couldbe used to keep sea clocks in precise agreement (Yoder 1990), Huygens carried out a series ofexperiments in an efort to understand the phenomenon. He found that synchronization did notoccur when the clocks were removed at a distance or oscillated in mutually perpendicular planes.Huygens deduced that the crucial interaction came from very small movements of the commonframe supporting the two clocks. He also provided a physical explanation for how the framemotion set up the anti-phase motion, but though his prowess was great his tools were limited: his ti t th ti h ti b t th h hi t hi t l li it d hidiscovery of synchronization occurred in the same year when young Isaac Newton removed tohis country home to escape the Black Plague, and begin the work that eventually led to hisPrincipia published some 20 years later The Royal Society viewed Huygens s explanation ofPrincipia, later. Huygens’ssynchronization as a setback for using pendulum clocks to determine longitude at sea (Birch1756). Occasion was taken here by some of the members to doubt the exactness of the motion ofthese watches at sea, since so slight and almost insensible motion was able to cause an alteration seain their going. Ultimately, the innovation of the pendulum clock did notsolve the longitude problem (Britten 1973). However, Huygens’ssynchronization observations have served to inspire study of sympatheticrhythms of interacting nonlinear oscillators in many areas of science.
  • 23. Huygens experimentThe pendulum in each clock measured ca. 9 in. in length, correspondingto an oscillation period of ca. 1 s. Each pendulum weighed 1/2 lb. andregulated the clock through a verge escapement, which required eachpendulum to execute large angular displacement amplitudes of ca. 20o ormore from vertical for the clock to function for a detailed description ofthe verge escapement). The amplitude dependence of the period in theseclocks was typically corrected by use of cycloidal-shaped boundaries toconfine th suspension (Huygens 1986). E h pendulum clock was fi the i (H 1986) Each d l l kenclosed in a 4 ft{ long case; a weight of ca. 100 lb was placed at thebottom of each case (to keep the clock oriented aboard a ship.)
  • 24. An original drawing of Huygens illustrating his experiments with pendulum clocks
  • 25. M. Bennett et al. (2002) Proc.M B lR. Soc. Lond. A (2002) 458, 563-579
  • 26. Van der Pol oscillator ( ) && + d y y 2 − 1 y + k y y = 0 my &
  • 27. Two coupled clocksmi l 2ϕ i + mi &&l cos ϕ i + cϕi miϕ i + mi gl sin ϕ i = M Di mi , && x &⎛ ⎞ ( ) 2 2⎜ M + ∑ mi ⎟ && + cx x + k x x + ∑ mi l ϕi cos ϕi − ϕi2 sin ϕi = 0, x & && &⎝ i =1 ⎠ i =1
  • 28. Parameters Pendulum Beamm1=1.0 [kg], M=10.0 [kg],m2=controling parameter, cx=1.53 [Ns/m],l=g/4π2=0.2485 [m], kx=3.94 [N/m],cϕ1=0.0083×m1 [N ] 0 0083× [Ns],cϕ2=0.0083×m2 [Ns], Escapement mechanism γN=5.0o MN1=0.075×m1 [Nm] MN2=0.075×m2 [Nm]
  • 29. Assuming the small amplitudes of the pendulums’ oscillations (typically for pendulumclocks Φ<2π/36 and for clocks with long pendulums Φ is even smaller one can describethe pendulum’s motion in the following form: h d l ’ i i h f ll i f ϕ i = Φ i sin (αt + β i ), ϕ i = αΦ i cos(αt + β i ), & ϕ i = −α 2 Φ i sin (αt + β i ). && Substituting above eqs to equation of motion:⎛ ⎞ ( ) 2 2⎜ M + ∑ mi ⎟ && + c x x + k x x = ∑ mi lα 2 Φ i sin(αt + β i ) + mi lα 2 Φ i3 cos 2 (αt + β i ) sin(αt + β i ) . x &⎝ i =1 ⎠ i =1Considering cos 2 α sin α = 0.25 sin α + 0.25 sin 3α , we get: 2U = M + ∑ mi , F1i = mi lα 2 (Φ i + 0.25Φ i3 ), F3i = 0.25mi lα 2 Φ i3 , i =1 2U&& + c x x + k x x = ∑ (F1i sin(αt + β i ) + F3i sin(3αt + 3β i ) ). x & i ( i ( i =1
  • 30. Assuming the small value of the damping coefficient cx previous equation can berewritten in the following form 2 x = ∑ ( X 1i sin(αt + β i ) + X 3i sin(3αt + 3β i ) ), i =1 where: F1i mi lα 2 (Φ i + 0.25Φ 3 ) X 1i = = i , kx −α U2 kx −α U 2 F3i 0.25mi lα 2 Φ 3 X 3i = = i . k x − 9α U 2 k x − 9α U 2implies the following acceleration of the beam M p g 2 && = ∑ ( A1i sin(αt + β i ) + A3i sin(3αt + 3β i ) ), x i =1 mi lα 4 (Φ i + 0.25Φ 3 ) A1i = − i , kx −α U 2 0.25mi lα 4 Φ 3 A3i = − i . k x − 9α U 2
  • 31. Energy balanceThe work done by the escapement mechanism during tone period of pendulum’soscillations can be expressed as T γN Wi DRIV = ∫ M Diϕi dt = 2 ∫ M Ni dϕi = 2M Niγ N . & 0 0Energy dissipated in the damper is given byE di i t d i th d i i b T T Wi DAMP = ∫ cϕiϕ dt = ∫ cϕiα 2 Φ i2 cos 2 (αt + β i )dt = παcϕi Φ i2 . & i 2 0 0 The energy transferred from the i-th pendulum to the beam M (pendulum looses part of its energy to force the beam to oscillate), so we have: t f it t f th b t ill t ) h T Wi SYN = ∫ mi &&l cosϕ iϕ i dt. x & 0 Energy balance for the i-th pendulum Wi DRIV = Wi DAMP + Wi SYN .
  • 32. Energy balance during the anti-phase synchronization (identical pendulums)In the case of the anti-phase synchronization oftwo identical pendulums the beam M is in rest(Czolczynski et al., 2009(a,b)). There is noenergy transfer between pendulums Wi DRIV = Wi DAMP . 2 M Ni γ N = παcϕi Φ i2 2 M Niγ N Φi = . παcϕi
  • 33. Energy balance - non-identical pendulumsSetting β1=0.0 (one of the phase angles can be arbitrarily chosen) and linearizingpendulum motion m1l 2α 4πΦ1 W SYN =− m2 Φ 2 sin β 2 = W SYN , k x − α 2U 1 m2 l 2α 4πΦ 2 W SYN = m1Φ1 sin β 2 = −W SYN . k x − α 2U 2Both synchronization energies are equal and the energy balance of bothpendulums have following form: d l h f ll i f W1DRIV = W1DAMP + W SYN W2DRIV + W SYN = W2DAMP
  • 34. Energy balance - non-identical pendulumsFinally we get: y g m1l 2α 4πΦ1 2 M N 1γ N = παcϕ1Φ − 2 m2 Φ 2 sin β 2 , Φi 0.9 kx − α U 1 2 0.8 0.7 0.6 m2l α πΦ 2 2 4 2 M N 2γ N = παcϕ 2 Φ 2 + m1Φ1 sin β 2 , 0.5 k x − α 2U 2 0.4 0.3 0.2 0.1We get 2 equations, as a parameter we take sin(β), 0 β2 0 1 2 3 4 5 6 7we plot angles as a function of parameter sin(β) p g p (and then we numerically find the phase shift β .
  • 35. Energy balance of pendulums; (a) anti-phase synchronization of identicalppendulums – there is no transfer of energy between p gy pendulums, ( ) p , (b) phasesynchronization of the pendulums with different masses: m1=1.0 [kg] and m2=0.289 [kg] and Φ1≈γN=5.0o – pendulum 1 transfer energy to the pendulum 2 via the beam M.
  • 36. Analytically we can find condition of in-phase andphase synchronization for both cases: identical andnon-identical masses of pendulums pendulums. But this not the end of the story…
  • 37. Parameters Pendulum Beamm1=1.0 [kg], M=10.0 [kg],m2=controling parameter, cx=1.53 [Ns/m],l=g/4π2=0.2485 [m], kx=3.94 [N/m],cϕ1=0.0083×m1 [N ] 0 0083× [Ns],cϕ2=0.0083×m2 [Ns], Escapement mechanism γN=5.0o MN1=0.075×m1 [Nm] MN2=0.075×m2 [Nm]
  • 38. Identical clocks
  • 39. Nonidenticalclocks
  • 40. Decreasing mass of second pendulum
  • 41. Increasing mass of second pendulum Comparison with analytics results
  • 42. How harmonic are clocks? m1=1 0 [kg]; =1.0 m2=11.0 [kg]
  • 43. Long period synchronization ϕ
  • 44. Long period synchronization
  • 45. Escapement mechanism – are the parameters important ?γNi − maximum angle below which escapement mechanism generate momentMNi - constant moment of escapement mechanism Assumption: ∫M Di dϕ i = M Ni γ Ni = Const
  • 46. Basin of attraction for, d ee t different sets of escapement o escape e t mechanism parametersInitialI iti l conditions: diti x(0) = 0, x(0) = 0.0, & ϕ i 0 = Φ sin β i 0 ,ϕ i 0 = αΦ cos β i 0 . &Parameters: γN =4.8 o ( ) 4 8 (a); γN =4.9o (b); γN =5.0o (c); γN =5 05o (d); =5.05 γN =5.1o (e); γN =5.2o (f).
  • 47. Poincare maps forchosen attractors γN =4.8o, T 23 (a), 4 8 T=23 ( ) γN =4.9o, T=6 (b), γN =4.9o, T=11 (c), γN =5 0o, T=11 (d); =5.0
  • 48. Poincare maps forchosen attractors γN =5.0o, chaos ( ) 50 h (e), γN =5.1o, T=59 (f), γN =5.2o, T=13 (g), γN =5 2o, T=35 (h); =5.2
  • 49. Rare attractors• Blekhman, I., and Kuznetsova, L. "Rare events - rare attractors; formalization andexamples", Vibromechanika, Journal of Vibroengineering, 10, 418-420 (2008)• Z k h k M S h ki I and Y Zakrzhevsky, M., Schukin, I. d Yevstignejev V "R i j V. "Rare attractors i d i in driven nonlinear lisystems with several degree of freedom", Sci. Proc. Riga Tech. Uni. 6(24), 79-93, (2007)• Chudzik, A., P. P., A. S. and T. K.: "Multistability and rare attractors in van der Pol -Duffing oscillator", International Journal Bifurcation and Chaos (2011), accepted forpublication
  • 50. Definition – open problem Our proposal
  • 51. As an example of the system whichpossesses multistability and rareattractors we consider an externallyexited van der Pol-Duffing oscillatorwhere: α=0.2, F=1.0, ω=0.955.For simplicity set of accessibleparameters is followingSets of possible initial conditions:
  • 52. • Long Period Synchronization• Multistability• Sensitivity on escapement mechanism parameters• Rare attractors e cos• Chaos
  • 53. More Clocks ?
  • 54. Possible configurations• the complete synchronization in which all p pendula behave identically, y,• pendula create three or five clusters of synchronized pendula pendula,• anti-phase synchronization in pairs (for even n and identical clocks),• uncorrelated behavior of all pendula
  • 55. Energy extraction from the sea waves
  • 56. Thank you !